Questions — SPS (686 questions)

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SPS SPS FM 2021 November Q8
11 marks Standard +0.3
In this question you must show all stages of your working. The function \(f\) is defined by \(f(x) = (1 + 2x)^{\frac{1}{2}}\).
  1. Find \(f'''(x)\) (i.e. the third derivative of \(f\)) showing all your intermediate steps. Hence, find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term. [8 marks]
  2. Use the expansion of \(e^x\) together with the result in part (a) to show that, up to and including the \(x^3\) term, $$e^x(1 + 2x)^{\frac{1}{2}} = 1 + 2x + x^2 + kx^3,$$ where \(k\) is a rational number to be found. [3 marks]
SPS SPS FM 2021 November Q9
7 marks Standard +0.3
  1. Show that $$\frac{1}{9r - 4} - \frac{1}{9r + 5} = \frac{9}{(9r - 4)(9r + 5)}$$ [2 marks]
  2. Hence use the method of differences to find $$\sum_{r=1}^{n} \frac{1}{(9r - 4)(9r + 5)}.$$ [5 marks]
SPS SPS FM 2021 November Q10
13 marks Challenging +1.8
\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3\sqrt{\cos(2\theta)}, \quad \text{where } -\frac{\pi}{4} < \theta \leq \frac{\pi}{4}, \quad \frac{3\pi}{4} < \theta \leq \frac{5\pi}{4}$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.
  1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4 marks]
  2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9 marks]
SPS SPS SM 2021 November Q1
8 marks Moderate -0.8
Find \(\frac{dy}{dx}\) for the following functions, simplifying your answers as far as possible.
  1. \(y = \cos x - 2 \sin 2x\) [2]
  2. \(y = \frac{1}{2}x^4 + 2x^4 \ln x\) [3]
  3. \(y = \frac{2e^{3x} - 1}{3e^{3x} - 1}\) [3]
SPS SPS SM 2021 November Q2
6 marks Moderate -0.3
  1. Express \(\frac{5x+7}{(x+3)(x+1)^2}\) in partial fractions. In this question you must show all of your algebraic steps clearly. [3] The function \(f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}\) can be written in the form; $$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
  2. Hence find the exact value of \(\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx\) [3]
SPS SPS SM 2021 November Q3
5 marks Standard +0.3
In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = (1 - 3x)(3 - x)^3$$ [5]
SPS SPS SM 2021 November Q4
5 marks Standard +0.3
Find the equation of the normal to the curve \(y = 4 \ln(2x - 3)\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(ax + by + k = 0\) where \(a > 0\). [5]
SPS SPS SM 2021 November Q5
4 marks Moderate -0.3
  1. Write \(\log_{16} y - \log_{16} x\) as a single logarithm. [1]
  2. Solve the simultaneous equations, giving your answers in an exact form. $$\log_3 y = \log_3(9 - 6x) + 1$$ $$\log_{16} y - \log_{16} x = \frac{1}{4}$$ [3]
SPS SPS SM 2021 November Q6
7 marks Challenging +1.2
  1. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$(\cos x + \sin x)(\cos x - \sec x) \equiv 2 \cot 2x$$ [3]
  2. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin\left(2x + \frac{\pi}{6}\right) = \frac{1}{2}\sin\left(2x - \frac{\pi}{6}\right)$$ [4]
SPS SPS SM 2021 November Q7
5 marks Standard +0.3
The diagram below represents the graph of the function \(y = (2x - 5)^4 - 1\) \includegraphics{figure_7}
  1. Find the intersections of this graph with the \(x\) axis. [1]
  2. Hence find the exact value of the area bounded by the curve and the \(x\) axis. [4]
SPS SPS SM 2021 November Q8
11 marks Standard +0.3
  1. Express \(2\sqrt{3} \cos 2x - 6 \sin 2x\) in the form \(R\cos(2x + \alpha)\) where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) [3]
  2. Hence
    1. Solve the equation \(2\sqrt{3} \cos 2x - 6 \sin 2x = 6\) for \(0 \leq x \leq 2\pi\) Giving your answers in terms of \(\pi\). [3]
  3. It can be shown that \(y = 9 \sin 2x + 4 \cos 2x\) can be written as \(y = \sqrt{97} \sin(2x + 24.0°)\)
    1. State the transformations in the order of occurrence which transform the curve \(y = 9 \sin 2x + 4 \cos 2x\) to the curve \(y = \sin x\) [3]
    2. Find the exact maximum and minimum values of the function; $$f(x) = \frac{1}{11 - 9 \sin 2x - 4 \cos 2x}$$ [2]
SPS SPS SM 2021 November Q9
7 marks Moderate -0.3
    1. Show that \(\cos^2 x \equiv \frac{1}{2} + \frac{1}{2}\cos 2x\) [1]
    2. Hence find \(\int 2\cos^2 4x \, dx\) [3]
  2. Find \(\int \sin^3 x \, dx\) [3]
SPS SPS SM 2021 November Q10
7 marks Standard +0.3
  1. The parametric equations of a curve are \(x = \theta \cos \theta\) and \(y = \sin \theta\) Find the gradient of the curve at the point for which \(\theta = \pi\) [3]
  2. A curve is defined parametrically by the equations; $$x = \cos \theta \qquad y = \left(\frac{\sin \theta}{2}\right)\left(\sin \frac{\theta}{2}\right)$$ Show that the cartesian equation of the curve can be written as \(y^2 = \frac{1}{8}(1-x)^2(1+x)\) [4]
SPS SPS SM 2022 October Q1
2 marks Easy -1.8
Simplify \(\left(\frac{x^{12}}{16}\right)^{-\frac{3}{4}}\) [2]
SPS SPS SM 2022 October Q2
5 marks Easy -1.3
A curve \(C\) has equation \(y = f(x)\) where $$f(x) = -3x^2 + 12x + 8$$
  1. Write \(f(x)\) in the form $$a(x + b)^2 + c$$ where \(a\), \(b\) and \(c\) are constants to be found. [3]
The curve \(C\) has a maximum turning point at \(M\).
  1. Find the coordinates of \(M\). [2]
SPS SPS SM 2022 October Q3
5 marks Moderate -0.8
In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Simplify $$\frac{\sqrt{32} + \sqrt{18}}{3 + \sqrt{2}}$$ giving your answer in the form \(b\sqrt{2} + c\), where \(b\) and \(c\) are integers. [5]
SPS SPS SM 2022 October Q4
6 marks Moderate -0.3
The equation $$(k + 3)x^2 + 6x + k = 5$$, where \(k\) is a constant, has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k^2 - 2k - 24 < 0$$ [4]
  2. Hence find the set of possible values of \(k\). [2]
SPS SPS SM 2022 October Q5
7 marks Moderate -0.8
  1. Given that $$y = \log_3 x$$ find expressions in terms of \(y\) for
    1. \(\log_3\left(\frac{x}{9}\right)\)
    2. \(\log_3 \sqrt{x}\)
    Write each answer in its simplest form. [3]
  2. Hence or otherwise solve $$2\log_3\left(\frac{x}{9}\right) - \log_3 \sqrt{x} = 2$$ [4]
SPS SPS SM 2022 October Q6
6 marks Moderate -0.8
An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54
  1. show that $$a + 4d = 6$$ [2]
Given also that the 8th term is half the 7th term,
  1. find the values of \(a\) and \(d\). [4]
SPS SPS SM 2022 October Q7
6 marks Moderate -0.3
In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Using algebra, find all solutions of the equation $$3x^3 - 17x^2 - 6x = 0$$ [3]
  2. Hence find all real solutions of $$3(y - 2)^6 - 17(y - 2)^4 - 6(y - 2)^2 = 0$$ [3]
SPS SPS SM 2022 October Q8
7 marks Standard +0.3
\includegraphics{figure_2} The resting heart rate, \(h\), of a mammal, measured in beats per minute, is modelled by the equation $$h = pm^q$$ where \(p\) and \(q\) are constants and \(m\) is the mass of the mammal measured in kg. Figure 2 illustrates the linear relationship between \(\log_{10} h\) and \(\log_{10} m\) The line meets the vertical \(\log_{10} h\) axis at 2.25 and has a gradient of \(-0.235\)
  1. Find, to 3 significant figures, the value of \(p\) and the value of \(q\). [3]
A particular mammal has a mass of 5kg and a resting heart rate of 119 beats per minute.
  1. Comment on the suitability of the model for this mammal. [3]
  2. With reference to the model, interpret the value of the constant \(p\). [1]
SPS SPS SM 2022 October Q9
7 marks Challenging +1.2
A sequence of numbers \(a_1, a_2, a_3, \ldots\) is defined by $$a_{n+1} = \frac{k(a_n + 2)}{a_n}$$, \(n \in \mathbb{N}\) where \(k\) is a constant. Given that
  • the sequence is a periodic sequence of order 3
  • \(a_1 = 2\)
  1. show that $$k^2 + k - 2 = 0$$ [3]
  2. For this sequence explain why \(k \neq 1\) [1]
  3. Find the value of $$\sum_{r=1}^{80} a_r$$ [3]
SPS SPS SM 2022 October Q10
7 marks Standard +0.3
A circle \(C\) with radius \(r\)
  • lies only in the 1st quadrant
  • touches the \(x\)-axis and touches the \(y\)-axis
The line \(l\) has equation \(2x + y = 12\)
  1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]
Given also that \(l\) is a tangent to \(C\),
  1. find the two possible values of \(r\), giving your answers as fully simplified surds. [4]
SPS SPS FM 2023 January Q1
5 marks Easy -1.3
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 2 & a \\ 0 & 1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 2 & a \\ 4 & 1 \end{pmatrix}\). \(\mathbf{I}\) denotes the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf{A} + 3\mathbf{B} - 4\mathbf{I}\). [3]
  2. \(\mathbf{AB}\). [2]
SPS SPS FM 2023 January Q2
4 marks Moderate -0.3
The transformations \(\mathbf{R}\), \(\mathbf{S}\) and \(\mathbf{T}\) are defined as follows. \begin{align} \mathbf{R} &: \quad \text{reflection in the } x\text{-axis}
\mathbf{S} &: \quad \text{stretch in the } x\text{-direction with scale factor } 3
\mathbf{T} &: \quad \text{translation in the positive } x\text{-direction by } 4 \text{ units} \end{align}
  1. The curve \(y = \ln x\) is transformed by \(\mathbf{R}\) followed by \(\mathbf{T}\). Find the equation of the resulting curve. [2]
  2. Find, in terms of \(\mathbf{S}\) and \(\mathbf{T}\), a sequence of transformations that transforms the curve \(y = x^3\) to the curve \(y = \left(\frac{1}{3}x - 4\right)^3\). You should make clear the order of the transformations. [2]